Method for sizing a rotor with a non-through shaft, associated rotor and motor-compressor set

ABSTRACT

The rotor for an electrical machine with a non-through shaft intended to drive a transmission line comprises two half-shafts enclosing a cylindrical magnetic mass.The magnetic mass comprises at least two adjacent identical cells, the cells being configured to prevent the propagation of metadamping in the rotor over a range of excitation frequencies of the transmission line, the rotation frequency range of the transmission line being hypercritical.

FIELD OF THE INVENTION

The present invention relates to mechanical systems comprising a rotorwithout a through shaft.

The present invention also relates to a motor-compressor set comprisingsuch a rotor.

BACKGROUND

FIG. 1 illustrates an example of a motor-compressor set 1 according tothe prior art including a rotary electrical machine 2 connected to acompression section 3.

The rotary electrical machine 2 comprises a stator 4 wherein there isinserted a rotor 5 with a through shaft including a shaft 5 a passingthrough magnetic plates 5 b and driving the compression section 3 at arotation speed Ω.

The compression section 3 comprises compression wheels 6 mounted on ashaft 7 of the section 3.

The shaft 7 of the section 3 is connected to the shaft 5 a of the rotor5.

Bearings 8 and 9 maintain the rotor shaft 5 a and the shaft 7 of thesection 3 in rotation on an axis A.

The bearings 8 and 9 comprise for example magnetic bearings or fluidbearings such as oil, water or gas bearings.

A fluid is generally injected into the casing of the motor-compressorset in order in particular to cool the rotary electrical machine 2, forexample a gas or a liquid.

In a variant, the casing of the motor-compressor set 1 is empty.

The bearings 8 and 9 preferably include magnetic bearings generallyhaving a control cutoff frequency of 5.5 kHz.

Generally, in operation, the rotation speed of the rotor shaft 5 a isdetermined so that the transmission line, comprising the rotor 5 and theshaft 7, does not pass through the first direct-precession bending mode,the transmission line not operating in a hypercritical speed domain.

Operation of the transmission line in a hypercritical speed domaincauses a loss of stability of the system in rotation comprising themotor 2 and the compression section 3 that may result in the destructionthereof.

The hypercritical transmission shaft line comprises several bendingresonant frequencies (eigenmodes) between the first direct-precessionbending eigenmode and the control cutoff frequency of the magneticbearings 8 and 9 that may dynamically deform the magnetic mass, controlof the magnetic bearings 8 and 9 exciting said eigenmodes.

This loss of stability is generated by the appearance of metadampingsappearing to a preponderant extent in the plates 5 b of the laminatedrotor 5 when the transmission line exceeds the critical speed.

The overall efficiency of the motor-compression set 1 does in particulardepend on the rotation speed of the shaft 7.

This limitation of the usable-speed domain degrades the efficiency ofthe motor-compressor set 1.

FIG. 2 illustrates a schematic dynamic model of the transmission line ofthe motor-compressor set 1 in a fixed reference frame (X, Y, Z).

The rotor 5 is represented by its mass M_(R), its stiffness K_(R), itsinternal damping D_(R), a coefficient α representing the gyroscopiceffects applied to the shaft 5 a and proportional to the polar moment ofinertia of the rotor 5, and its center of gravity G.

The fluid injected into the casing of the motor-compressor set 1 ismodeled by a mass M_(C), a damping coefficient D_(C) and a stiffnessK_(C).

The rotor of the compression section 3 comprising the shaft 7 and thewheels 6 is represented by its stiffness K_(DISP) and its mass M_(DISP).

The pressure forces exerted on the wheels 6 are ignored, the compressionsection being designed to minimize them.

The bearings 8 and 9 are represented by stiffness coefficients K_(BY)along the axis Y and K_(BZ) along the axis Z, and damping coefficientsD_(BY) along the axis Y and D_(BZ) along the axis Z.

Let M_(L) be the mass of the whole of the transmission line equal to thesum of the mass M_(R) and of the mass M_(DISP), and K_(L) the stiffnessof the transmission line equal to the sum of the stiffness K_(R) and ofthe stiffness K_(DISP).

According to the general engineering equations, the instability speedΩ_(INST) of the transmission line is:

$\begin{matrix}{\Omega_{INST} = \left\{ {{\frac{K_{Y} + K_{Z}}{2 \cdot \left( {M_{L} + M_{C} - \alpha} \right)} \cdot \ \left( {1 + \frac{D_{C} + \frac{D_{BY} + D_{BZ}}{2}}{D_{R}}} \right)^{2}} + \frac{\left( {K_{Y} - K_{Z}} \right)^{2}}{4 \cdot D_{R}^{2}}} \right\}^{1/2}} & (1) \\{{{where}\mspace{14mu} K_{Y}} = \frac{1}{\frac{1}{K_{BY}} + \frac{1}{K_{L}} + \frac{1}{K_{C}}}} & (2) \\{{{and}\mspace{14mu} K_{Z}} = \frac{1}{\frac{1}{K_{BZ}} + \frac{1}{K_{L}} + \frac{1}{K_{C}}}} & (3)\end{matrix}$

In order to increase the instability speed Ω_(INST), a first solutionconsists of modifying the bearings 8 and 9 so as to increase thestiffnesses K_(BY) and K_(BZ) and/or to increase the dampings D_(BY) andD_(BZ).

However, it is difficult to modify the characteristics of the bearings,in particular in the case of magnetic bearings having identicalstiffnesses K_(BY) and K_(BZ) that are very small compared with thethickness K_(L), and equal dampings D_(BY) et D_(BZ).

A second solution for increasing the instability speed Ω_(INST) consistsin increasing the stiffness K_(L) of the transmission line or reducingthe mass M_(L) of the transmission line.

However, the transmission line is sized to transmit a sufficientmechanical power between the electrical machine 2 and the compressionsection 3, and modifying at least one characteristic of the transmissionline may cause damage thereto.

A third solution for increasing the instability speed Ω_(INST) consistsof acting on the internal damping D_(R) of the rotor 5 dependent on therotation speed of the rotor 5, by minimizing the internal damping D_(R),that is to say by pushing back the frequency domain where metadampingsappear.

The document US 2013/0025961 discloses that a structure comprising aperiodic arrangement of cells forming the structure can increasemetadampings in a predetermined range of frequencies.

Furthermore, as the rotary electrical machine 2 comprises the rotor 5with through shaft 5 a, the peripheral speed of the rotor is limited to200 m/s in order to limit the concentration of stresses in the magneticplates 5 b generated under the effect of the centrifugal force andliable to damage the rotor. This limitation of the rotation speeddegrades the efficiency and the compression ratio of themotor-compressor set 1.

SUMMARY

It is therefore proposed to overcome all or some of the drawbacks of themechanical systems driven by a rotary electrical machine according tothe prior art, in particular by sizing the rotor so that the rotaryelectrical machine turns at a rotation frequency higher than thehypercritical rotation frequency of the transmission line whileremaining stable.

In the light of the above, a rotor is proposed for an electric machinewith a non-through shaft intended to drive a transmission line,comprising two half-shafts enclosing a cylindrical magnetic mass.

The magnetic mass comprises at least two adjacent identical cells, thecells being configured to prevent the propagation of metadamping in therotor over a frequency range of excitation of the transmission line, therotation frequency range of the transmission line being hypercritical.

According to one feature, each cell comprises at least one circularelement.

Advantageously, each cell includes at least two circular elements.

Preferably, the two circular elements comprise identical or differentoutside diameters.

According to another feature, the two circular elements are made fromidentical materials or each plate being made from a different material.

Advantageously, each circular element is produced from magnetic steel,copper or plastics material.

According to yet another feature, the rotor furthermore comprises twoshort-circuit disks and two conductive bars housed in the circularelements in order to form a squirrel cage.

Preferably, the hypercritical rotation frequency range extends from 0 to500 Hz.

According to another aspect, a rotary electrical machine comprising therotor as defined above is proposed.

Preferably, the rotary electrical machine is of the squirrel-cageasynchronous type.

According to yet another aspect, a mechanical system is proposedcomprising the rotary electrical machine as defined above and amechanical device comprising a transmission shaft, the transmissionshaft being connected to one of the rotor half-shafts.

According to another aspect, a motor-compressor set is proposed,comprising at least one compression section, and the rotary electricalmachine as defined above and driving said compression section.

According to yet another aspect, a method for sizing a rotor with anon-through shaft for an electric machine driving a transmission line isproposed.

The method comprises a sizing of a cell so as to prevent the propagationof metadamping in the rotor over a range of excitation frequencies ofthe transmission line, the rotation frequency range of the transmissionline being hypercritical, the rotor comprising two half-shafts enclosinga cylindrical magnetic mass comprising at least two adjacent identicalcells.

According to one feature, the hypercritical rotation frequency rangeextends from 0 to 500 Hz.

Preferably, the sizing of the cell including at least one circularelement is done analytically using a beam model.

Other features and advantages of the invention will emerge from areading of the following description of embodiments of the invention,given solely by way of non-limitative examples and with reference to thedrawings disclosed herewith.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1 and 2, already mentioned, illustrate a motor-compressor setaccording to the prior art;

FIG. 3 illustrates an embodiment of a motor-compressor set;

FIG. 4 illustrates an example of a cell; and

FIG. 5 shows an example of a curve representing the homogenized Rayleighdamping coefficient.

DETAILED DESCRIPTION

As used throughout, the term “figure” is equivalent to FIG., which is anacronym for Figure.

Reference is made to FIG. 3, which illustrates a cross section in anaxial direction of an example embodiment of a motor-compressor set 10comprising a rotary electrical machine 11 connected to a compressionsection 12.

The rotary electrical machine 11 comprises a rotor 13 comprising anon-through shaft of central axis B connected to a transmission shaft 14of the compression section 12.

The rotor 13 and the transmission shaft 14 form a transmission line.

It is supposed hereinafter that the rotation frequency of thetransmission line is situated in the hypercritical frequency domain ofoperation of the motor-compressor set 1 and is less than 500 Hz, thehypercritical frequency domain of the hypercritical transmission lineall being the rotation frequencies higher than or equal to the firstdirect-precession bending resonant frequency of the transmission line.

The diameter of the transmission shaft 14 is sized according to thevalue of a torque to be transmitted.

The rotor 13 and the transmission shaft 14 are maintained in rotation bytwo bearings 15 and 16 situated respectively at the free end of therotor 13 and of the transmission shaft 14.

The bearings 15 and 16 are for example bearings on an oil film, on a gasfilm, with magnetic levitation, or fluid bearings.

Generally, the bearings 15 and 16 comprise magnetic bearings.

The rotor 13 is inserted in a stator 17 of the rotary electrical machine11 of the squirrel-cage asynchronous type.

In a variant, the rotary electrical machine 11 may be a machine of thewound-rotor asynchronous type or of the synchronous type with a woundrotor or with permanent magnets.

The rotor 13 with a non-through shaft comprises a cylindrical magneticmass 18 enclosed between a first 19 and a second 20 half-shaft by meansof two short-circuit disks 21 and 22, one end of the transmission shaft14 being connected to the second half-shaft 20.

The magnet mass 18 comprises adjacent identical cells 23 to 31, each ofthe cells having a total length L.

The magnetic mass 18 comprises a periodic repetition of cells 23 to 31.

In a variant, the magnetic mass 18 may comprise a non-periodicrepetition of cells 23 to 31.

The cells 23 to 31 are configured to prevent the propagation ofmetadamping in the rotor 13 over a range of excitation frequencies ofthe transmission line.

The excitation frequency range is for example between 0 and 5.5 kHz.

By placing the metadamping frequency range of the magnetic mass 18 ofthe rotor 13 beyond 5.5 kHz, the eigenmodes of the transmission line arenot excited by the control of the magnetic bearings 15 and 16.

Each cell 23 to 31 comprises two circular elements 32 and 33respectively with different outside diameters D32 and D33 andrespectively with different thicknesses L32 and L33.

The circular elements 32 and 33 are produced from different materials,the first plate 32 being produced from a first material characterized bya Young's modulus E₃₂ and a density ρ₃₂, and the second plate 33 beingproduced from a second material characterized by a Young's modulus E₃₃and a density ρ₃₃.

Naturally, each cell 23 to 31 may comprise more than two circularelements.

The circular elements are for example produced from magnetic steel inorder to form magnetic plates, from copper, from non-metallic materialor from plastics material.

Furthermore, the adjacent circular elements may be separated from eachother by varnish, a resin, an electrical insulator, a layer of paint, aceramic deposit or any polymer.

The circular elements 32 and 33 forming adjacent magnetic plates may beseparated by an electrical insulator.

In a variant, the outside diameters D32 and D33 may be identical.

According to another variant, the thicknesses L32 and L33 may beidentical.

According to yet another variant, the first and second materials may beidentical.

According to another embodiment, each cell 23 to 31 may comprise a stackof metal sheets, the thickness of the metal sheets preferably beinggreater than 5% of the outside diameter of the magnetic mass 18.

Conductive bars 34 are housed in the circular elements 32 and 33 and theshort-circuit disks 21 and 22 so that the short-circuit disks 21 and 22and the conductive bars 34 form a squirrel cage.

Tie rods 35 are distributed uniformly over a diameter D of the magneticmass 18 so as to maintain the circular elements 32, 33 compacted betweenthe half-shafts 19 and 20.

The compression section 12 comprises compression wheels 36 mounted onthe transmission shaft 14 so that the rotor 13 rotates the wheels 36 inorder to compress a gas or a fluid.

As the rotor 13 has a non-through shaft, the peripheral speed of therotor 13 is not limited to 200 m/s, making it possible to improve theefficiency of the rotary electrical machine 11. The higher the rotationspeed of the rotor 13, the greater the power developed by the rotaryelectrical machine 11.

More precisely, the motor-compressor set 10 can rotate at ahypercritical rotation frequency.

A direct reference frame R1(X, Y, Z) is defined, comprising an axis Xcoincident with the rotation axis of the rotor 13.

The calculation of the damping D_(R13) of the rotor 13 is detailedhereinafter.

The rotor 13 is modeled by a beam of length L extending along the axis Xin the reference frame XYZ, of cross section A.

As the rotation frequency of the rotor 13 is less than 500 Hz, it isaccepted that the deformations of the cross section of the beam due toshear are ignored and that the rotation inertia effect is ignored.

The behavior under bending of the beam according to time t and theposition x in the beam along the axis X when no force is applied to therotor 13 is governed by:

$\begin{matrix}{{{{E(x)}{I(x)}\frac{\partial^{4}{y\left( {x,t} \right)}}{\partial x^{4}}} + {{\rho(x)}{A(x)}\frac{\partial^{2}{y\left( {x,t} \right)}}{\partial t^{2}}}} = 0} & (4)\end{matrix}$

where

E(x) is the Young's modulus of the magnetic mass 18 equal to E₃₂ or E₃₃according to the value of x, I(x) is the moment of inertia of the crosssection along the axis x of the beam, and ρ(x) is the density equal toρ₃₂ or ρ₃₃ according to the value of x, and A(x) is the cross section ofthe rotor 13 according to the value of x.

Then the variables x and t are separated by introducing the complexfunction:

y(x, t)=Y(x)e ^(−jωt)  (5)

where j²=−1

The Young's modulus E(x) is replaced by a complex Young's modulus Ē(x)and the density ρ(x) by a complex density ρ(x) such that:

E (x)=E(x)(1−jη _(E)·ω)  (6)

where η_(E) is the damping factor of the Young's modulus, and

$\begin{matrix}{\overset{\_}{\rho}{(x) = {{\rho(x)}\left( {1 - {j\frac{\eta_{\rho}}{\omega}}} \right)}}} & (7)\end{matrix}$

where η_(ρ) is the damping factor of the density.

The solution of equation (4) is of the type:

Y ₁(x)=(A ₁ e ^(z1.x) +B ₁ e ^(z2.x) +C ₁ e ^(z3.x) +D ₁ e ^(z4.x))e^(−jωt) for x being in the first plate 32  (8), and

Y ₂(x)=(A ₂ e ^(s1.x) +B ₂ e ^(s2.x) C ₂ e ^(s3.x) +D ₂ e ^(s4.x))e^(−jωt) for x being in the second plate 33  (9),

the coefficients z1 to z4 and s1 to s4 being the roots of equation (4).

As the magnetic mass 18 comprises a periodic repetition of identicalcells 23 to 31 of period L=L₃₂+L₃₃, the Floquet-Bloch theorem is appliedto equations (8) and (9):

Y ₁(x)=(A ₁ e ^(z1.(x-L)) +B ₁ e ^(z2.(x-L)) +C ₁ e ^(z3.(x-L)) +D ₁ e^(z4.(x-L)))e ^(−jωt) e ^(−kL) for x being in the first plate 32  (10),and

Y ₂(x)=(A ₂ e ^(s1.(x-L)) +B ₂ e ^(s2.(x-L)) +C ₂ e ^(s3.(x-L)) +D ₂ e^(s4.(x-L)))e ^(−jωt) e ^(−kL) for x being in the second plate 33  (11),

where k is the complex Floquet wave number so as to be limited to one ofthe cells 23 to 31 shown in FIG. 4, equation (10) describing thedeflection of the magnetic mass 18 for the values of x less than 0 andequation (11) describing the deflection of the magnetic mass 18 for thevalues of x greater than 0, the origin of the axis X being chosen at theinterface of the first and second circular elements 32 and 33.

By symmetry, the coefficients z1 and s1, z2 and s2, z3 and s3, z4 and s4are equal.

By applying the conditions at the limits of continuity of the momentsand of the curvature effects, a matrix system of eight equations isobtained, comprising the eight coefficients A₁ to D₁ and A₂ to D₂ ofequations (10) and (11).

Let M(k,ω) be the matrix containing the matrix system for an angularfrequency ω1 of the frequency of rotation of the rotor 13.

The equation of the determinant DET of the matrix M equal to 0 is solvedso as to determine the four complex values of k denoted k 1, k 2, k 3and k 4.

Moreover, using equation (4), the following wave number is obtained:

$\begin{matrix}{{\overset{\_}{k} = {\sqrt{\omega}\left( \frac{\rho_{h}A_{h}}{E_{h}I_{h}} \right)^{\frac{1}{4}}\left( \frac{1 - {j\eta_{\rho}}}{1 - {j\eta_{E}}} \right)^{\frac{1}{4}}}}{where}} & (12) \\{{\rho_{h}A_{h}} = {\frac{1}{L}{\int_{0}^{L}{{\rho(x)}{A(x)}{dx}}}}} & (13)\end{matrix}$

ρ(x) being equal to ρ₃₂ if x is between −L₃₂ and 0, and to ρ₃₃ if x isbetween 0 and L₃₃, and

the cross-section A(x) being calculated from the diameter D₃₂ is x isbetween −L₃₂ and 0, and from the diameter D₃₃ is x is between 0 and L₃₃;

$\begin{matrix}{\frac{1}{E_{h}I_{h}} = {\frac{1}{L}{\int_{0}^{L}\frac{dx}{{E(x)}{I(x)}}}}} & (14)\end{matrix}$

E(x) being equal to E₃₂ if x is between −L₃₂ and 0, and to E₃₃ if x isbetween 0 and L₃₃, and

the moment of inertia I(x) being calculated from the diameter D₃₂ if xis between −L₃₂ and 0, and from the diameter D₃₃ if x is between 0 andL₃₃.

Let η_(h) be the homogenized Rayleigh damping coefficient representingthe damping of the rotor 13.

$\begin{matrix}{\eta_{h} = {\frac{\eta_{\rho}}{\omega} + {\eta_{E} \cdot \omega}}} & (15)\end{matrix}$

Using the values k 1, k 2, k 3 and k 4 determined during the calculationof the determinant DET of the matrix M(k,χ) for the angular frequency ω1and equation (12) for to equal to ω1, the following equations areobtained, making it possible to calculate the coefficient η_(h) for theangular frequency to ω1:

Re( ki)=F1(η_(ρ), η_(E))  (16)

Im( ki)=F2(η_(ρ), η_(E))  (17)

i varying from 1 to 4

where Re is the real part of ki and Im is the imaginary part of ki.

The coefficient η_(h) the angular frequency ω1 varying from 0 rad/s to34557 rad/s (i.e. 5.5 kHz) is calculated for various values of L₃₂, L₃₃,D₃₂, D₃₃, E₃₂, E₃₃, ρ₃₂ and ρ₃₃.

Following a variational analysis the amplitude of the metadampings inthe frequency domain considered is the smallest when each cell 23 to 31comprises a single plate.

FIG. 5 shows a curve C1 representing the value of the diameter D₃₂ as afunction of the thickness L₃₂, each cell 23 to 31 including only theplate 32.

The pairs of values (D₃₂, L₃₂) situated in the zone Z1 above the curveC1 generate metadampings in the rotor 13 for frequencies below 5.5 kHz,and the pairs of values (D₃₂, L₃₂) situated in the zone Z2 under thecurve C1 generate, in the rotor 13, metadampings the amplitudes of whichare not liable to cause instability of the rotor 13 for frequenciesabove 5.5 kHz.

For example, a rotor 13 comprising cells 23 to 31 each including a platehaving a Young's modulus E₃₂, a density ρ₃₂, a thickness L₃₂₁ and adiameter D₃₂₁ is not unstable in the frequency domain ranging from 0 to5.5 kHz if, for the value of the diameter D₃₂ fixed at D₃₂₁, the valueof L₃₂ is less than L321, otherwise the rotor 13 is unstable.

The method described above makes it possible to control the vibratorybehavior of the rotor 13 in order to minimize the metadampings due tothe rotating parts that may cause instability at the hypercriticalrotation frequencies and thus determine a range of rotation frequenciesminimizing metadampings.

The transmission chain driven by the rotor 13 functions in ahypercritical rotation frequency domain without presenting any risk ofinstability.

Functioning in the hypercritical rotation frequency domain makes itpossible to increase the overall efficiency and the compression ratio ofthe motor-compressor set 10.

In a variant, the rotor 13 may be connected to a mechanical deviceconfigured to drive the rotor 13 so that the rotary electrical machine11 functions in generator mode in order to produce electric power, theassembly comprising said device and the electric machine functioning ata hypercritical rotation frequency.

As the electric power produced is proportional to the rotation speed ofthe rotor of said machine, the efficiency of said machine is improved.

Naturally, the sizing of the rotor 13 as defined above, and so that itfunctions at a hypercritical rotation frequency of the transmissionline, applies for any mechanical system comprising a transmission shaftconnected to the rotor.

1. Rotor for an electrical machine with a non-through shaft intended todrive a transmission line, comprising two half-shafts enclosing acylindrical magnetic mass, characterized in that the magnetic masscomprises at least two adjacent identical cells, the cells beingconfigured to prevent propagation of metadamping in the rotor over arange of excitation frequencies of the transmission line, the range ofrotation frequencies of the transmission line being hypercritical. 2.Rotor according to claim 1, wherein each cell includes at least onecircular element.
 3. Rotor according to claim 1, wherein each cellincludes at least two circular elements.
 4. Rotor according to claim 3,wherein the two circular elements comprise identical or differentoutside diameters.
 5. Rotor according to claim 3, wherein the twocircular elements are made from identical materials or each circularelement is made from a different material.
 6. Rotor according to claim2, wherein each circular element is produced from magnetic steel, copperor plastics material.
 7. Rotor according to claim 1, further comprisingtwo short-circuit discs and conductive bars housed in the circularelements in order to form a squirrel cage.
 8. Rotor according to claim1, wherein the hypercritical rotation frequency range extends from 0 to500 Hz.
 9. Rotary electrical machine comprising the rotor according toclaim
 1. 10. Rotary electrical machine according to claim 9, of theasynchronous type with squirrel cage.
 11. Mechanical system comprisingthe rotary electrical machine according to claim 9 and a mechanicaldevice comprising a transmission shaft, the transmission shaft beingconnected to one of the rotor half-shafts.
 12. Motor-compressor setcomprising at least one compression section and the rotary electricalmachine according to claim 9 driving said compression section. 13.Method for sizing a rotor with non-through shaft for an electricalmachine driving a transmission line, characterized in that the methodcomprises a sizing of a cell so as to prevent the propagation ofmetadamping in the rotor over a range of excitation frequencies of thetransmission line, the rotation frequency range of the transmission linebeing hypercritical, the rotor comprising two half-shafts enclosing acylindrical magnetic mass comprising at least two adjacent identicalcells.
 14. Method according to one of claims 13, wherein thehypercritical rotation frequency range extends from 0 to 500 Hz. 15.Method according to claim 13, wherein the sizing of the cell includingat least one circular element is done analytically using a beam model.